The inventory literature has received attention in manufacturing
industry almost a century ago. In 1913, Ford Harris suggested an idea of
how many parts to make in a batch. The Harris order batching model has
then been forgotten until the idea was later published by Wilson (1934)
in Harvard Business Review. The batching rule then became known as the
Wilson Economic Order Quantity (EOQ) as it applies to inventory
management.

The model assumes that a retailer of a product buys the product at
a constant unit cost, incurs a fixed cost per order, stores the product
at a constant carrying cost per unit of inventory per year, and faces a
deterministic and constant demand rate over an infinite horizon, the
retailer's optimal strategy is to buy a fixed quantity every time
he or she replenishes the inventory. Ignoring inventory related costs,
classical price theory tells us that when a product's demand is
price sensitive but the demand curve is known and stationary, the
retailer's optimal strategy is to charge a single price throughout
the year. Although Whitin (1955) was the first one to integrate the
concepts of inventory theory with the concepts of price theory to
investigate the simultaneous determination of price and order quantity
decisions of a retailer, he never stated so explicitly. Whitin's
(1955) model would adhere to all the assumptions of the EOQ model stated
above, except that demand is price sensitive, with a known and
stationary demand curve, a retailer's optimal strategy would be,
once again, to buy a fixed quantity for every inventory cycle and to
sell it at a single price.

Kunreuther and Richard(1971) then showed that when demand is price
elastic, centralized decision-making (using simultaneous determination
of optimal price and order quantity) was superior to the common practice
of decentralized decision-making whereby the pricing decisions were made
by the marketing department while the order quantity decisions were made
by the purchasing department independently. Although Kunreuther and
Richard (1971) were perhaps unaware of Whitin's (1955) paper, their
model was very similar to Whitin's (1955) model. Assuming a known
and stationary demand curve along with the remaining conditions of the
EOQ model, Arcelus et al (1987, page 173) asserted: "given constant
marginal costs of holding and purchasing the goods, the firm will want
to maintain the same price throughout the year". Again, they
assumed a fixed single selling price throughout each inventory cycle.
What they did not realize is that, even though marginal holding costs
are constant per unit, a firm's holding costs at any particular
time within an inventory cycle are a function of inventory on hand,
which itself is a function of the time from the beginning of the
inventory cycle.

Since Whitin's (1955) work, numerous authors (Tersine &
Price,1981; Arcelus & Srinivasan,1987; Ardalan, 1991; Hall, 1992;
Martin, 1994; Arcelus & Srinivasan, 1998; Abad, 2003) have used
Whitin's (1955) and Kunreuther and Richard's(1971) models as
foundations to their own models. But none of these authors have ever
questioned Whitin's (1955) and Kunreuther and Richard's (1971)
assumption that the retailer's optimal strategy would be to sell
the product at a fixed price throughout the inventory cycle. The fact
that Whitin's (1955) and Kunreuther and Richard's (1971)
assumption of a single price throughout an inventory cycle leads to
suboptimal profits for the retailer is due to declining carrying costs
as a function of time. However, any optimization model allowing a
retailer with a price-insensitive demand to set the selling price
arbitrarily would push the price to infinity. In other words, in that
situation, price is not seen as a decision variable for any mathematical
model. Given an arbitrary price (and corresponding demand), the
retailer's only strategy is to minimize his inventory ordering and
holding costs by using the EOQ model.

Considering a situation of price sensitive demand, Abad (1997;
2003) found that, in the case of a temporary sale with a forward buying
opportunity, a retailer's optimal strategy is to charge two
different prices during the last inventory cycle of the quantity bought
on sale--a low price at the beginning of the inventory cycle and a
higher price starting somewhere in the middle of the cycle. Yet, Abad
(1997; 2003) did not consider a similar strategy in every regular
inventory cycle of a product with price sensitive demand. Inspired by
Abad's (1997; 2003), Joglekar et al (2008) showed that a
continuously increasing price strategy: charging a relatively low
selling price at the beginning of an inventory cycle when the on-hand
inventory is large, would lead to higher profit.

With the widespread use of revenue management or yield management
techniques (Feng & Xiao, 2000; McGill &. van Ryzin,1999; Smith,
Leimkuhler & Darrow, 1992; Talluri & van Ryzin,2000; Weatherford
&. Bodily, 1992. in the airline, car rental, and hotel industries
today, a time-dependent (or dynamic) pricing strategy has become
commonly adopted. Revenue management techniques are typically applied in
situations of fixed, perishable capacity and a possibility of market
segmentation (Talluri & van Ryzin, 2000). In recent years, retail
and other industries have begun to use dynamic pricing policies in view
of their inventory considerations. The recent recession has brought
forth dynamic pricing to a new light. As retail sales dropped,
businesses were facing unusual built-ups of inventory that would lead to
order cancellations affecting all parties of the supply chain.

In this paper, we extend the dynamic pricing model to incorporate
the gradual usage (or gradual production/ replenishing) inventory. (See
Chase et al, 2010) for the gradual usage model.) Consider a retailer
that is also a manufacturer of a product. Depending on the demand and
the ordering quantity, the retailer would produce the inventory
gradually until it fulfills the planned order quantity. Since the
build-up is gradual, the actual inventory holding cost will be lowered
and that the maximum inventory level will not be as high as the basic
EOQ. In light of the currently changing demands with economic
conditions, this model is appropriate when the retailer/manufacturer
would have to adjust his or her inventory policy and production schedule
according to the current price elasticity of demand dictated by the
market.

Although this model is limited by the fact that it assumes
deterministic customer behavior (or deterministic price elasticity) and
any lack of competitive reactions to one's action, it does provide
the ground work to take on a new direction of inventory management in
supply chain. In addition, it provides a way to reset initial pricing to
optimize profit, and the marginal rate of price increase during an
inventory cycle. The most beneficial of it all is that the solutions are
only dependent on the assumed price elasticity and cost data, and can be
revised easily should the market conditions change again.

In what follows, we first recapitulate Whitin's (1955) and
Kunreuther and Richard's (1971) fixed price strategy model with
gradual usage cycle as described in Chase et al (2010). In the next
section, we present our own model. The derived model has a polynomial
objective function, and hence, does not have a closed form solution.
However, we are still able to provide the optimal initial pricing and
the marginal rate of price increase during the cycle easily. Using
Microsoft Solver[R], we utilize several numerical examples with a linear
demand curve and varied values of relevant parameters. The final section
provides the conclusions of our analysis along with some directions for
future research.

THE FIXED PRICE GRADUAL USAGE MODEL

Both papers of Whitin (1955), Kunreuther and Richard (1971)
consider a situation where all the other assumptions of the EOQ model
are valid but demand is price sensitive, with a known and stationary
demand curve. Whitin's (1955) notation is different from Kunreuther
and Richard's (1971) notation. There are also some minor
differences in the details of the two models. In addition with Chase et
al (2010) for gradual usage assumption, the following captures the
basics of these models. Although the model is applicable to any form of
the demand function, for simplicity, we use a linear demand function.
Let the following notations hold,

C = retailer's known and constant unit cost of buying the
product,

S = retailer's known and constant ordering cost per order,

I = retailer's carrying costs per dollar of inventory per
year,

[P.sub.1] = retailer's selling price per unit in this model,

m = constant production rate per period,

t = time elapsed from the beginning of an inventory cycle,

[Y.sub.1] = retailer's profit per cycle,

[Z.sub.1] = retailer's profit per period

It is assumed that,

[P.sub.1] > C, and [D.sub.1] = retailer's annual demand as
a function of the selling price, [P.sub.1], hence, [D.sub.1] = a - b
[P.sub.1] where a and b are nonnegative constants, a representing the
theoretical maximum annual demand (at the hypothetical price of $0 per
unit) and b representing the demand elasticity (i.e., the reduction in
annual demand per dollar increase in price). Although P1 would remain
constant throughout a cycle, we choose to express it in an affine
function form to be consistent with the price increasing model in the
next section.

Note that since [D.sub.1] must be positive for the conceivable
range of values of [P.sub.1], a > b[P.sub.1] for that range of values
of [P.sub.1], and since [P.sub.1] > C, it follows that a > bC.

Let

[T.sub.1] = the duration of retailer's gradual replenishment
in an inventory cycle, and

Q = retailer's order quantity per order in this model. Then, Q
= [D.sub.1][T.sub.2] = (a-b[P.sub.1])[T.sub.2]. The maximum inventory
level would then be m[T.sub.1]-[DT.sub.1] = (m-a+b[P.sub.1])[T.sub.1].

From [T.sub.1] [greater than or equal to] t [greater than or equal
to] [T.sub.2], the retailer would be consuming from the inventory at a
rate of [D.sub.1]t until the end of the cycle, [T.sub.2]. That is, we
have

Notice that Equation (5) is equivalent of the gradual usage (or
production) model as illustrated in

Chase et al (2010) .

Combining Equations (4) and (5) and simplifying, we have an
explicit quartic equation in terms of [T.sub.2]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

As we can see here, we can solve Equation (6) for [T.sub.2] which
has 4 roots. To obtain these roots, one may use MathCAD to find all the
closed form solutions in terms of the coefficients of [T.sub.2], and
substituting them into (3) to see which root would optimize [Z.sub.1].
On the other hand, one may use other software to solve (4) and (5)
simultaneously. However, we elect to use Excel Solver[R] to solve for
[P.sub.1] and [T.sub.2] simultaneously for optimal [Z.sub.1]. To ensure
solution quality, we have also implemented the constraints such as
[P.sub.1] > C, and so on. We then verify the solution with (4), (5)
and (6). Once care is taken to input these conditions and reasonable
starting values, in our experimentation, Excel Solver has never failed
to return the best real solution (if any). Hence, we believe that
practicing managers would be adequately served by the use of Excel
Solver. They need not worry about obtaining all the roots of the quartic
equation.

THE CONTINUOUSLY INCREASING PRICE GRADUAL USAGE MODEL

We retain all of the assumptions of the foregoing model, except
that now we assume that the retailer uses a continuously increasing
price strategy within each inventory cycle.

Let us add the following notation:

[P.sub.2](t) = the retailer's selling price at time t, and
[P.sub.2](t) = f+gt, where f and g are nonnegative decision variables,
and f > C. f represents the retail price at the beginning of each
inventory cycle and g represents the annual rate of increase in the
retail price throughout an inventory cycle.

Differentiating [Z.sub.2] with respect to [T.sub.2], f, and g, for
the first order optimal conditions by setting them to zero, we have,
respectively,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

From (14) and (15), we obtain,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

and,

g = (3/m)ICf(1-b)+1/2IC(17) (17)

Notice that f represents the initial price set at the beginning of
the inventory cycle, and the marginal increase during the cycle is given
by g as in Equation (17). Also note that when m approaches infinity as
in the case of basic EOQ models with instantaneous replenishment, g
would be equal to IC, same as the basic model.

Of course, when we replace f and g in Equation (13), we would
obtain an explicit polynomial function of [T.sub.2] , that is, of fifth
order, or quintic function. There is, unfortunately, no close form
solution for any polynomial function higher than quartic. For
practitioners, we would, once again, rely on Excel Solver which has
demonstrated to be a reliable tool for obtaining an optimal solution.

NUMERICAL EXAMPLE: BASE CASE

Consider a situation where the retailer's cost of a product is
$5 per unit. The theoretical maximum periodic demand is 20 units and
periodic demand declines at the rate of 1 unit for each dollar's
increase in the price. The ordering costs are $100 per order and the
carrying costs are $0.05 per dollar of inventory per period. The
production rate is 40 units per period. We will refer to this set of
assumptions as the base case.

Table 1 summarizes the base case assumptions, the optimal decisions
and the consequences under the two models. As can be seen there, in the
fixed price model, the optimal retail price is $15.64 per unit and the
optimal inventory cycle time is 10.35 periods. This means that the
retailer would order 45.088 units per order and would obtain per period
profit of $31.08.

In the continuously increasing price production model, the starting
retail price is $12.75 per unit at the beginning of the cycle and that
price increases at the rate of $0.125 per period. The optimal cycle time
is 12.118 periods. This means that the retail price at the end of the
cycle is $14.26 per unit, the retailer would order 69.548 units per
order and would achieve a periodic profit of $39.88.

In addition to reporting these numbers, Table 1 also presents the
percent differences between the two models for each relevant decision
and consequence. Observe that, in percent terms, the differences are
rather substantial. In comparison with the fixed price strategy, the
continuously increasing price strategy results in a longer cycle time of
17.08 percent. At the beginning of an inventory cycle, under the
continuously increasing price model, the retail price is smaller than
what it is under the fixed price model. However, by the end of the
cycle, the retail price under the continuously increasing price model is
still lower than what it is under the fixed price model. As a result,
the periodic demand is larger under the continuously increasing price
model. The average cycle profit shows a substantial improvement of 28.31
percent in the continuously increasing price model compared to the per
cycle profit under the fixed price model. This shows that continuously
pricing model performs substantially better than a fixed pricing model
by taking advantage of the gradual usage/replenishment, and the pricing
elasticity.

The specific numerical results we have obtained are a function of
the numerical assumptions we have made. Hence, in order to identify the
circumstances under which the continuously increasing price strategy
would be particularly desirable, we carried out a sensitivity analysis,
as described in the following section.

SENSITIVITY ANALYSIS

Table 2 summarizes the results of an analysis where we increased
the value of each one of our parameters (except the last row which
indicates a decrease in production level), one at a time, while
maintaining the values of the other parameters constant. In each case,
Table 2 shows the consequences of these changes on the retailer's
annual profits under the two models and the percentage increase in the
periodic profit that the retailer obtains by using the continuously
increasing price gradual usage strategy as against using the fixed price
strategy. For comparison purposes, the first row of Table 2 repeats the
profit results of the two models in the base case.

While other things remain constant, any increase in ordering cost,
elasticity, inventory holding rate, and production/replenishment rate
would still favor the continuously increasing price strategy. With the
exception of production/replenishment rate, the periodic profits can
increase rather significantly over those of the fixed price model,
although the differences are getting narrower. In particular, when the
production/replenishment rate is doubled, the profit difference is only
1.69%. This, we believe is due to the fact the demand rate, m, is due to
the fact that when increasing production/replenishment rate would be
equivalent to switching gradual production to instantaneous delivery of
the basic EOQ model. As Joglekar et al (2008) has pointed out that the
percentage changes would be small under the basic EOQ assumptions.

The additional analysis of decreasing the production/replenishment
rate, m, is shown in the last row of Table 2. Notice that the gradual
usage model discussed in Chase et al (2010) exhibits lower inventory
cost if m approaches the periodic demand. In the continuously increasing
price strategy, the profit increase is even more pronounced given the
fact that we would now take advantage of the slower replenishment, as
well as the price elasticity leading to a 23.96% increase with m = 20
over the fixed price model.

Of course, our sensitivity analysis focused on changes in one
parameter at a time. When several parameters are favorable to the
continuously increasing price strategy, the gains offered by this
strategy may be even more significant.

CONCLUSIONS

Traditionally, operations researchers (Whitin, 1955: Kunreuther
& Richard, 1971, and others) have assumed that when a product's
demand curve is known and stationary, a retailer of the product would
find it optimal to buy a fixed quantity every time he buys and to sell
the product at a fixed price throughout the year. Joglekar et al (2008)
found that the continuously increasing price strategy would increase the
periodic profit abide the small gains. With the gradual usage
assumption, Chase et al (2010) shows that by take advantage of the
noninstantaneous production/replenishment rate, the total inventory cost
could be reduced. However, when employing the continuously increasing
price strategy in this situation as proposed by Joglekar et al (2008),
we find that the profit per period would increase significantly as
indicated in the numerical examples in the previous sections.

A continuously increasing price strategy might be impractical in
the past, but with the new technology today, e-tailers can easily update
their prices continuously. Elmaghraby and Keskinocak's (2003)
review of dynamic price models indicates that a number of industries are
already using continuously changing pricing strategies.

With the recent economic downturn, retailers are more conscientious
of their inventory and pricing. The ever changing retail environment may
warrant a new and more dynamic strategy in order to remain competitive
in the market place. Our model provides a practical and systematic
approach to coordinate a supply chain's sales and productions. In
addition, it provides an easy computational tool of the initial price
and the marginal rate of increase. Should the business environments
change; a new pricing scheme can be quickly re-configured and
implemented just like what we have shown with our formulas in section 3.

Our numerical example suggests that the advantage of a continuously
increasing price strategy is significant and the sensitivity analysis
shows that this strategy is particularly desirable when demand is highly
price sensitive or when an e-tailer's supplier commands great
pricing power. While the continuously increasing price strategy may not
be practical for a brick-and-mortar retailer, such a retailer could use
the dual price strategy model developed by Joglekar (2003) that a
retailer who sets two different prices at two different points in an
inventory cycle obtains a greater profit than a retailer using a single
fixed price throughout the cycle. Although Joglekar et al (2008) model
only shows modest gains under the basic EOQ assumptions, the
continuously increasing price strategy with gradual usage (or
non-instantaneous replenishment) provides significantly higher profits.
In addition, it provides an avenue where retailers can adjust their
pricing schemes in a hurry when the market conditions call for sudden
and significant changes in price elasticity.

There are several directions in which this research can be further
investigated. In this paper, we have extended the Joglekar et al (2008)
model for a retailer who is also a manufacturer of the product. Some
retailers have now adopted the pre-ordering strategy (such as game
console, smart phones, and so on), which is similar to back-ordering.
This extension would do well for those with different pre-order and
initial prices to maximize profit. Finally, a number of recent models
coordinating pricing and order quantity decisions across a supply chain
[23-26] have assumed a fixed price for each participant of the supply
chain. These models also need to be updated by considering a
continuously increasing price strategy.

ACKNOWLEDGMENT

This research is dedicated to Professor Prafulla Joglekar who
passed away in November 2009.

Boyaci, T. & G. Gallego, (2002). Coordinating pricing and
inventory replenishment policies for one wholesaler and one or more
geographically dispersed retailers, International Journal of Production
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Harris, F. (1913) How many parts to make at once, Factory, The
Magazine of Management, 10(2),.

Joglekar, P. , (2003). Optimal price and order quantity strategies
for the reseller of a product with price sensitive demand, Proceedings
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